The overall reaction that occurs when sugar is metabolized is $$ \mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}(s)+12 \mathrm{O}_{2}(g) \longrightarrow 12 \mathrm{CO}_{2}(g)+11 \mathrm{H}_{2} \mathrm{O}(l) $$ For this reaction, \(\Delta H^{\circ}\) is \(-5650 \mathrm{~kJ}\) and \(\Delta G^{\circ}\) is \(-5790 \mathrm{~kJ}\) at \(25^{\circ} \mathrm{C}\). (a) If \(25 \%\) of the free energy change is actually converted to useful work, how many kilojoules of work are obtained when one gram of sugar is metabolized at body temperature, \(37^{\circ} \mathrm{C}\) ? (b) How many grams of sugar would a 120 -lb woman have to eat to get the energy to climb the Jungfrau in the Alps, which is \(4158 \mathrm{~m}\) high? \(\left(w=9.79 \times 10^{-3} \mathrm{mh}\right.\), where \(w=\) work in kilojoules, \(m\) is body mass in kilograms, and \(h\) is height in meters.)

When studying the metabolism of sugar, it's essential to understand the concept of free energy change, denoted as \(\Delta G\). This term refers to the amount of energy that is available to do work during a chemical reaction, once the energy that is lost to entropy is subtracted. In simpler terms, it's the usable energy that can be harnessed from the reactions occurring within our bodies.

Free energy change is a key indicator of the spontaneity of a reaction—negative \(\Delta G\) values suggest that the reaction can occur spontaneously, meaning no additional energy is required to drive it forward. In the exercise provided, the free energy change for the metabolism of sugar is \(\Delta G^\circ = -5790 \, \text{kJ}\). The negative sign indicates that the metabolism of sugar is a spontaneous process, releasing energy.

When we consider a real-world scenario, such as the human body, not all the energy from the metabolism of sugar is converted into work. In fact, a portion of it is lost as heat due to the second law of thermodynamics. The exercise mentioned that only 25% of this energy actually becomes useful work. Taking this into account is crucial for practical applications, like calculating the number of calories a person may need to climb a mountain, as the body's efficiency in converting food energy to mechanical work is limited.

The metabolism of sugar is a process that falls under the broader field of thermochemistry, which investigates the heat involved in chemical reactions. One of the central concepts in thermochemistry is enthalpy change (\(\Delta H\)), representing the total heat change in a system at constant pressure.

In our context, the \(\Delta H^\circ\) for the metabolism of sugar is given as \(\Delta H^\circ = -5650 \text{kJ}\). This indicates an exothermic reaction wherein energy is released in the form of heat. This heat release is a byproduct of the reaction and contributes to the total energy metabolism of organisms.

Understanding the enthalpy change is significant for calculating the energy content of foods, determining dietary requirements, and even assessing the potential impacts of different food sources on bodily functions. The thermochemical principles enable us to correlate the biochemical processes occurring in the body with the energy manifested as heat.

Stoichiometry is the aspect of chemistry that relates to determining the relative quantities of reactants and products in chemical reactions. In the context of sugar metabolism, stoichiometry provides the framework for quantifying the conversion of sugar molecules into energy and other byproducts.

From the stoichiometric equation given in the exercise, one can see that each molecule of sugar (\(\text{C}_{12}\text{H}_{22}\text{O}_{11}\)) reacts with twelve molecules of oxygen (\(\text{O}_2\)) to produce twelve molecules of carbon dioxide (\(\text{CO}_2\)) and eleven molecules of water (\(\text{H}_2\text{O}\)). This balanced equation helps us understand the proportions in which substances react and the proportions in which they produce the end products.

In practical applications, stoichiometry is applied to calculate the amount of a reactant needed to produce a desired amount of product. For instance, the exercise uses stoichiometry to determine the amount of sugar a woman must consume to supply enough energy for climbing a mountain. The calculation involves using the molar mass of sugar to find the number of moles of sugar corresponding to a given mass and subsequently using the free energy change to calculate the energy available from metabolizing those moles of sugar.